Cubicity

In graph theory, cubicity is a graph invariant defined to be the smallest dimension such that a graph can be realized as an intersection graph of unit cubes in Euclidean space.

Cubicity was introduced by Fred S. Roberts in 1969 along with a related invariant called boxicity that considers the smallest dimension needed to represent a graph as an intersection graph of axis-parallel rectangles in Euclidean space.

can be realized as an intersection graph of axis-parallel unit cubes in

-dimensional Euclidean space.

[2] The cubicity of a graph is closely related to the boxicity of a graph, denoted

The definition of boxicity is essentially the same as cubicity, except in terms of using axis-parallel rectangles instead of cubes.

Since a cube is a special case of a rectangle, the cubicity of a graph is always an upper bound for the boxicity of a graph.

In the other direction, it can be shown that for any graph

is the ceiling function, i.e., the smallest integer greater than or equal to

A cubicity 2 graph realized as the intersection graph of unit cubes, i.e. squares, in the plane.